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Intro to Power Lab Concepts
Created by the University of Illinois at Urbana-Champaign
TCIPG PMU Research Group1
Table of Contents
1. PRE-LAB
DC Power----------------------------------------------------------------------------------- 2-4
AC Power----------------------------------------------------------------------------------- 4-5
RLC Circuits/Phasors---------------------------------------------------------------------- 6-13
2. EQUIPMENT LIST------------------------------------------------------------------------------ 14
3. PROCEDURE------------------------------------------------------------------------------------ 15-17
Resistor Only Circuit
RC Circuit
RL Circuit
RLC Circuit
4. LAB SETUP HELP----------------------------------------------------------------------------- 18-20
5. OSCILLOSCOPE HELP----------------------------------------------------------------------- 21
6. POWER FLOW DEMO------------------------------------------------------------------------ 22-31
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Credit given to Mark Alikpala, Kenta Kirihara, Michael Quinlan, Bogdan Pinte, Andy Keehoon Yoon
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1. PRE-LAB:
Objective: This lab has been created so you can understand the fundamentals necessary for the
use of the power lab.
By the end of this lab, you will:
1. Understand what RMS is.
2. Know how capacitors and inductors affect phase difference between voltage and current.
3. Understand how phase difference affects power factor.
4. Differentiate between leading and lagging voltage based on power factor.
5. Know how to take Voltage, Current, and Power measurements using the multimeter,
oscilloscope, and power meter.
This pre-lab will review and introduce you the basic knowledge to conduct the lab in this order:
A. DC Power
B. AC Power
C. RLC Circuits/Phasors
A. DC Power
Shown below is a basic circuit with a DC (direct current) source and a resistor. We have a 10Ω
resistor connected to a 10V DC voltage source.
To find the current across the resistor, we use Ohm’s Law: V=IR. We have a 10V voltage source
and a 10Ω resistor, so our current in our diagram is 1A. This current is flowing from the positive
end of the voltage source to the negative end of the voltage source around the circuit, or in our
case clockwise around the circuit. If you recall this from your introductory physics classes, this
should be just a review. Now, to find the power that is consumed by this resistor, we use the
power equation P=VI. Since our current is 1A and our voltage across the terminal is 10V, our
power is 10W.
Next, observe the circuit below.
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We have two resistors in series with each other. Components are in series when the negative end
of one component is connected to the positive end of another. When two resistors are in series
with each other, we can simply add the resistance together to form one equivalent resistor so that
Rseries=R1+R2+....Rn. In our case, since each resistor is 10Ω, we can see that the equivalent
resistance is 20Ω. When we solve for current across both resistors, we use the equivalent
resistance; thus, our current is I=V/R=10/20=0.5A. Furthermore, the power dissipated totally by
both resistors is 5W.
Moving on, below is a different type of connection.
This type of connection is called a parallel connection. Components are in parallel when the
positive ends of the components are connected together and the negative ends of the components
are connected together. When resistors are connected in parallel, the equivalent resistance can
be found this way: 1/Rparallel=1/R1+1/R2+....1/Rn. We use this equation, and we get the
equivalent resistance as 5Ω. Thus, our current here is 10/5=2A. Consequently, our total power is
20W.
Now that we know how to take resistors in series and parallel and form an equivalent resistor,
we can do similar things for inductors as well. Inductors, usually denoted as L in the honor of
Lenz, can be added the same exact way as resistors: Lseries=L1+L2..Ln and 1/Lparallel=1/
L1+1/L2..1/Ln. However, capacitors, noted as C in circuits, are added in the opposite way as it
in resistors and inductors: 1/Cseries=1/C1+1/C2+....1/Cn and Cparallel=C1+C2+...Cn. Do not
worry about the current and power across these two yet.
B. AC Power
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Since we discussed DC sources, we will now discuss AC (Alternating Current) sources. AC
sources are sources that often use sine waves or square waves with a set amplitude. The general
form of AC sources, whether it be voltage or current, is sin(wt) where w=2
f=
angular frequency.
Look at the graph below.
Here is a sine wave with amplitude 1 and angular frequency (w) is 1. The equation for this is
sin(t). Let us say this is the equation for an AC voltage source V(t)=sin(t)V. The amplitude of
this voltage, as stated above, is 1. This means that the voltage ranges from -1V to 1V from time
to time. In other words, if we were to simply measure the voltage from the source, we will get
a time varying value. If we take the amplitude of this function and double it, we get a type of
reading called Peak-to-Peak. However, more commonly used is what we call a RMS(Root
Mean Square) reading. The general way to solve for RMS is shown below.
In sine waves, RMS=Peak-to-Peak/(
) =Amplitude/(
).
Essentially, this is what RMS does in graphical form. We take the graph from above, take the
absolute value of it, as shown below, and take the time average of it.
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So, what is the purpose of using RMS? Well, RMS eliminates the time factor involved in AC.
There is a difference between average and RMS. If you look at the graph previously referenced,
the time average voltage would be zero. As a matter of fact, any sine wave without a DC offset
would have a time average of zero. For example, if we take the RMS of V(t)=sin(t)V, we get 1/
(
)
Summary DC vs AC graph
C. RLC Circuits (R-resistance, L-inductance, C-capacitance)/Phasors
First, we will introduce three very important notions:
1. POWER FACTOR(PF) is defined as the cosine of the angle between current and
voltage.
PF=cos
where =
If the current is leading the voltage, our power meter will denote it as negative.
If the current is lagging, our power meter will denote it as positive.
2. REAL POWER(P)
P=Vrms*Irms*PF [W]
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This is the general power formula. P=V*I for DC circuits is a particular case of the
general power formula because Vrms=V and Irms=I for constant values of voltage and
current. Also, PF(in DC) is 1, therefore P=VI for DC circuits.
3. POWER TRIANGLE
Using P and PF, we can create power triangles as follows:
The top triangle shows a lagging power factor, while the triangle on the left is drawn for a
leading power factor, by definition.
These triangles will show up in RL and RC circuits because capacitors and inductors provide
reactive power. Resistor only circuits do not have reactive power, but these will be further
investigated in the different cases below:
●
Resistor only
The following circuit consists of an AC voltage source connected across a single
resistor:
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Note that the current and voltage in the capacitor are in phase, meaning that their peaks occur at
the same time. The only difference is in their amplitudes: current amplitude is smaller by a factor
of R. Also, since I=V/R, as the resistance increases, the amplitude of current decreases.
In phasor notation, the current and voltage are drawn as follows:
This simply shows that the current and voltage are in phase (
with phasor magnitude representing the amplitude.
apart)
In this case:
therefore PF=cos(0)=1 =>
P=Vrms*Irms [W]
Also, since
, the power triangle will only consist of
a horizontal line representing Real Power. Resistors DO
NOT have Reactive Power.
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Capacitor only
What happens when an alternating current is applied to a capacitor? Capacitor takes time
to charge, and discharge. Because of this, there is a phase shift between current and voltage.
More precisely, the current leads the voltage by 90 degrees (1/4 cycle).
●
One important property of a capacitor is its reactance, with the formula given below as it
depends on the value of the capacitor in farads as well as the frequency of the AC waveform:
Reactance is given by the symbol X and is also measured in ohms. The larger the
capacitance C, the more charge has to flow to build up a particular voltage on the plates, and the
higher the current will be. The higher the frequency of the voltage, the shorter the time available
to change the voltage, so the larger the current has to be. The current, then, increases as the
capacitance and the frequency increase.
In this case:
therefore PF=cos(-90)=0 => P=0 [W]
Since
, the power triangle will only consist of a vertical line pointing down (in
accordance to leading power triangle definition) representing Reactive Power. Capacitors DO
NOT have Real Power. Also, remember that adding capacitors in parallel increases C, therefore
the reactance(1/ C) decreases. So, as you add more capacitors in parallel the amplitude of the
Reactive Power decreases.
●
Inductor only
What happens when an alternating current is applied to an inductor? Inductors also take
time to charge, and discharge. Because of this, there is a phase shift between current and voltage.
More precisely, the current lags the voltage by 90 degrees (1/4 cycle).
The formula for inductive reactance is given by:
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Since Ohm’s law can also be written as I=V/
,the larger the inductance L and the
lower current will be. The current, then, decreases as the inductance and the frequency increase.
In this case:
therefore PF=cos(90)=0 => P=0 [W]
Since
, the power triangle will only consist of a vertical line pointing upwards (in
accordance to lagging power triangle definition) representing Reactive Power. Inductors DO
NOT have Real Power. Also, remember that adding Inductors in series increases L, therefore the
reactance( ) increases. So, as you add more inductors in series the amplitude of the Reactive
Power increases.
● RC circuit
Consider the following circuit, where an AC voltage source is connected across a series RC
circuit:
Notice that the phase shift is less than -90 degrees (current still leads voltage), which is because
the resistor is trying to bring the voltage and current in phase, while the capacitor is trying to
keep them 90 degrees out of phase; so they settle somewhere in between, depending on the given
values of R and C.
It follows that PF=cos(phase shift) and P=Vrms*Irms*PF. Note that PF will be negative by
definition.
Then, a power triangle can be drawn. Remember that the Real Power phasor corresponding
to resistance R is horizontal, while the Reactive Power phasor corresponding to capacitor C is
vertical pointing down:
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It is very important to understand what will happen to the
phase difference as more resistors and capacitors are
added in parallel (since the resistors in the resistor box
are wired in parallel, as well as the capacitors).
So, as you add resistors in parallel, the total resistance
goes down, increases the phase difference. However, as
you add capacitors in parallel, the capacitance C
increases and decreases the reactance
, which in turn,
decreases the phase difference.
RL circuit
Consider the following circuit, where an AC voltage source is connected across a series RL
circuit:
●
Notice that the phase shift is less than 90 degrees (voltage still leads current), which is because
the resistor is trying to bring the voltage and current in phase, while the inductor is trying to
keep them 90 degrees out of phase; so they settle somewhere in between, depending on the given
values of R and L.
It follows that PF=cos(phase shift) and P=Vrms*Irms*PF. Note that PF will be positive by
definition.
Then, a power triangle can be drawn. Remember that the Real Power phasor corresponding
to resistance R is horizontal, while the Reactive Power phasor corresponding to inductor L is
vertical pointing up:
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So, as you add resistors in parallel, the total resistance goes down and increases the phase
difference. However, as you add inductors in series, the inductance L increases, which increases
the reactance
, which in turn, increases the phase difference.
RLC circuit
Consider the following circuit, where an AC voltage source is connected across a series RLC
circuit:
●
The same formulas for PF and P are used as in the last two examples.
However, the PF can be either negative, positive or zero, depending on the resultant vector
addition of capacitive and inductive Reactive Powers.
Below is shown an example where the PF is positive(lagging). This means, as shown in the
triangle, that capacitive reactance(green) is smaller than inductive reactance(red), making the
overall reactance inductive.
A special case is when the capacitive and inductive reactances cancel each other. In this case,
there will only be Real Power (a horizontal phasor). Based on RL and RC circuits described
above, you should be able to tell what happens as more resistors and capacitors are added in
parallel, as well as inductors in series.
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Why is power factor important?
The power factor amplitude can take a value between 0 and 1. PF of 0 means there is only
reactive power (which can be either inductive or capacitive), while PF of 1 means there is
only Real Power. Ideally, it is good practice to have a PF closer to 1. This is because power
companies will charge for reactive power if it is above a certain limit.
How do you keep PF close to 1? Capacitors can be used to cancel out inductive reactance if there
is too much of it. On the other hand, if there is too much capacitive reactance, inductors should
be used to bring PF to 1.
Below is a concise summary of the power triangle. Make sure you fully understand it.
References:
http://www.eeweb.com/blog/andrew_carter/how-capacitors-behave-in-ac-circuits
http://people.sinclair.edu/nickreeder/eet155/mod04.htm
http://keywon.com/wiki/index.php?title=AC_Circuits#AC_Capacitor_Circuit
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2. EQUIPMENT LIST
Banana Cables
Voltage/Current Probe
Resistor Box
Capacitor Box
Inductor (42mH)
Power meter
Handheld multimeter
Variac
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3. PROCEDURE
PART A: Basic “R” Circuit.
1. Construct a basic circuit represented by the diagram below, and switch on one resistor
(refer to page X for lab setup help). Make sure the variac is turned off.
2. Connect the voltage probe to channel 1 and the current probe to channel 2 of the
oscilloscope to measure the current and voltage across the resistor.
3. Turn the master switch on, make sure you turn on the oscilloscope.
4. Turn the variac on and turn the knob until 50V from the power meter is read. Make sure
the readings from the handheld multimeter match.
a. Turn the knob to measure RMS Voltage and probe two nodes you want to
measure voltage across. (Resistor box in this case)
5. Measure RMS voltage, RMS current, and phase difference from the oscilloscope. (refer
to pg X for help)
a. First, you’ll have to initialize the equipment by pressing save/recall → recall
factory setup → ok
b. Make sure both waves are turned on by pressing CH 1 and CH2. You can verify
what channels are switched on by looking on the bottom-left of the screen.
c. Now, you’ll need to adjust the scale by using the knobs. Using scale knobs, rescale the graphs so they are easily observable. (each channel will have to be
scaled separately)
d. To measure different values, press measure and press more for appropriate
measurement.
e. For phase, make sure you’re measuring from Ch1 to Ch2. (negative phase should
be “leading” and positive should be “lagging”
f. If you see any fluctuations in the measurements, you can press Menu → Average
to “smooth out” the graphs.
6. Calculate PF and power from oscilloscope measurements.
a. Do you see any significant phase difference?
b. Do the calculated values match the power meter?
7. Switch on more resistors.
a. What changes do you see in current?
b. What changes do you see in power?
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8. After each part you can turn the variac switch off for safety reasons before you add more
components in the circuit.
PART B: RC Circuit
1. Add a capacitor box in series. Turn on one capacitor.
2. From the oscilloscope, measure phase difference, calculate the PF and verify the result
with the power meter.
3. From the measurements, determine real power, reactive power, and apparent power, then
draw the power triangle.
4. Try switching on more capacitors.
a. As you switch on more capacitors, does the phase angle decrease or increase?
Why? (Hint: How does the impedance affect the power triangle?)
PART C: RL Circuit
1. Construct an RL circuit with one resistor on.
2. From the oscilloscope, measure phase difference and calculate the PF and verify the
result with the power meter.
3. From the measurements, determine real power, reactive power, and apparent power, then
draw the power triangle.
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a. What difference do you see from RL circuit?
PART 4: RCL Circuit (Phase “Cancel”)
1. Add an inductor to the circuit in series and switch on one capacitor.
2. Measure the phase difference and calculate the PF and verify the result with the
multimeter.
3. Determine the real power, reactive power, and apparent power, then draw the power
triangle.
a. How much more reactive power should the capacitor draw in order to “cancel”
phase difference (in other words, make PF = 1)
b. What is the necessary capacitance? (Hint: try drawing out the impedance on the
complex plane)
4. Switch a few more capacitors on and see how the phase angle changes.
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4. LAB SETUP HELP
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5. OSCILLOSCOPE HELP
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6. POWER FLOW DEMO
Review:
From the previous exercises, we were able to demonstrate that we can plot the current or voltage waves
through an oscilloscope.
Figure 1: Oscilloscope Wave Output
You can also plot multiple waves simultaneously through an oscilloscope. By doing so, we can see we
can see if a wave is (a) in unity with another wave, (b) lagging another wave or (c) leading another wave.
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Figure 2a: Signal 1 and Signal 2 are in Unity
Figure 2b: Signal 1 Lagging Signal 2
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Figure 2c: Signal 1 Leading Signal 2
POWER FLOW
Picture the following set-up: Two machines are connected to its own shaft. On machine 1, the
shaft allows the motor to spin as fast as it wants. However, on machine 2, the shaft only allows the motor
to spin at a certain speed.
Figure 3: 2 Machines and its corresponding shafts
Initial scenario: Machine 2 is already running at a certain speed, and we set Machine 1 to run as fast
as machine 2 by rotating its shaft at a certain speed. The voltages on the terminals of Machine 1 and
Machine 2 (V1 and V2 respectively) are equal. We then flipped a switch to connect the Machine 1 and the
Machine 2 together. What will happen?
Result: The 2 machines will be “synchronized” together. The 2 machines will now be running at the same
speed. In the “synchronized” state, the 2 machines will run at the same speed despite changes in the shaft.
In this initial scenario, since the machines are running at the same speed, and there will be no
power flowing in between the two machines. If we look at their voltages on oscilloscope, we will see that
the 2 voltages are in unity.
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Figure 4: Unity Voltages and Power
Next Scenario: As discussed earlier, since both machines are now “synchronized”, they will both be
running at the same speed. What will happen if we rotate Shaft 1 faster?
Result: Since Machine 1 wants to rotate at the same speed, but is receiving more “power” to rotate from
shaft 1, it will pass the excess “power” from shaft 1 to machine 2. Machine 2 will now be receiving more
power from machine 1 which will cause it to rotate faster. To remedy this, shaft 2 will act as a break and
prevent it from happening.
Follow-up Question: How much power does shaft 2 require to act as a break to maintain Machine 2 to
run at a certain speed?
Answer: The shaft 2 will need the same amount of power passed from Machine 1. Therefore, the power
passed from Machine 1 will be used by shaft 2 to prevent Machine 2 to spin faster than the ‘synchronized’
state. In this case, Machine 1 will be acting as a “GENERATOR” and Machine 2 will be acting as
a “MOTOR/LOAD”.
Follow-up Question 2: What will happen to the phase difference between both machines?
Answer: The phase of Machine 1 will shift to the right, causing a phase difference between the 2 phase
angles of the voltages.
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Figure 5: Machine1 as generator, Machine2 as motor.
Next Scenario: What will happen if we rotate shaft 2 even faster (faster than earlier)
Result: More power will be delivered (and consequently, consumed) on shaft 2. The phase of Voltage 1
will shift to the right again, which will cause the phase difference between the two voltages to be greater.
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Figure 6: Machine1 as generator, Machine2 as motor, larger power flow
Next Scenario: Let’s return the shaft 1 “power” as it was before in synchronous state. What will happen?
Result: The phase voltage of Machine 1 will shift to the left, causing it to be in unity with Machine 2’s
voltage. There will be no power flowing across in between the two machines.
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Figure 7: Unity Voltages and Power (Again)
Next Scenario: Set Shaft 1’s “power” lower than it was before. What will happen?
Result: Shaft 1 will then require “power” to make Machine 1 spin faster. For this to happen, shaft 2 will
generate excess “power” and give it to shaft 2. The phase voltage of Machine 1 will shift to the left and
the power will flow from Machine 2 to Machine 1. Machine 2 will now act as a generator and Machine 1
will now act as a motor.
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Figure 8: Machine 1 as motor, Machine 2 as generator
Next Scenario: Set Shaft1’s “power” to be even lower. What will happen?
Result: More power will be delivered (and consequently, consumed) on shaft 1. The phase of Voltage 1
will shift to the lft again, which will cause the phase difference between the two voltages to be greater.
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Figure 9: Machine 1 as motor, Machine 2 as generator, larger power flow
Next Scenario: Let’s return to the original scenario with 0 “power” flowing between the two terminals.
Now, give the shaft1 significantly more “power” to spin 1 OR significantly reduce shaft 1’s power. What
will happen?
Result: There will be a system breakdown due to instantaneous large power demand. There will be no
power flowing in between the two machines and they will be “desynchronized”.
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Figure 9: System Breakdown
Conclusion: The Power Flow between the two machines heavily depended on the phase difference on its
voltages. Through observation, we determined that the power is (taking right power flowing to the right as
positive and power flowing to the left as negative):
0: Voltages are in phase.
Small +: Small Phase difference, Voltage 1 lagging Voltage 2.
Large + : Large Phase difference , Voltage 1 lagging Voltage 2.
Small -: Small Phase difference, Voltage 1 leading Voltage 2.
Large - : Large Phase difference, Voltage 1 leading Voltage 2.
This is because power flow is governed by the equation
Where
*NOTE: Abrupt demand of power can cause a system breakdown. A phase shift of over 90 degrees will
cause a system breakdown as well